Cylindrically-Symmetric Equilibria in Ideal MHD with Fractal Pressure Profiles

In ideal magnetohydrodynamics, unphysical, pressure-driven currents exist where flux surfaces with rational rotational transform coincide with pressure gradients, a situation Grad termed ``pathological'' [1]. As an alternative, we construct a~non-trivial, continuous~pressure profile that is flat on sufficiently wide intervals near each rational surface. Such a profile must be self-similar and thus fractal, because intervals of flat pressure exist around high-order rational surfaces at all scales. This infinite-resolution fractal pressure is analyzed as a homeomorphism of the Cantor set. Additionally, an algorithm has been written to numerically produce an approximation of the pressure profile, where only a finite number of rational surfaces are considered. Using this algorithm, we investigate the magnetic field and current profiles associated with the fractal pressure and a given rotational transform in cylindrical geometry. [1] H. Grad, Phys. Fluids 10 (1), 137 (1967).